with action and state weightsα > 0 andβ ≥ 0, respectively, and discount factor 0 < γ < 1. A larger return is obtained when, fromst, a low-probability actionatis performed and followed by a low-probability transition to a statest+1. Therefore, maximizing the return...
By the implicit function theorem we only need that I'(u)\neq 0 for all u\in \mathcal{M}. By contradiction suppose that I'(u)=0 for some u\in \mathcal{M}. Thus -\Delta u+(|x|^{\alpha -N}u^{2})u=\mu u^{q-1}+\frac{p-Nb}{q-Nb}u^{p-1}. (2.2) ...
Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow-up result is established by Pohozaev identity if dimension .doi:10.1002/mma.10219Ru WangZhisu LiuMathematical Methods in the Applied Sciences...
At first I will give complete definitions, state and prove simple lemmas, etc., in order to establish common language. As time progresses I expect that things will become a little more high-level, less self-contained, and more technical. Depending on your background and interests, you may ...
[70]. By the hyperbolicity of this limit cycle and the implicit function theorem [98], linear stability persists for small enough coupling parameters\(\{\lambda ,\vartheta \}\), which act as small perturbations of the decoupled system. Indeed, for small enough\(\lambda \), all fixed ...
Thus, eliminating x(s), the state vector, by combining the output equation and equation (5.50), the output vector y(s) is given by (5.51)y(s)=[C(sI−A)−1B+D]u(s)=G(s)u(s) where G(s) is called the transfer function matrix. In general, the transfer function matrix has...
When the exciting field is left continuous, the qubit final state describes a circle on the Bloch sphere, periodically reaching 0 and 1. The probability P1 of the qubit to be in 1 oscillates as a function of time t: (4)P1=sin2Ω2t. This phenomenon is called a Rabi oscillation. Such...
The implicit assumption is that (mild) solutions do actually exist, which will be proved later. Theorem 4.2 (Uniqueness and continuous dependence for the 1-2 junction). Let {{ϕr}3r=1, ¯f 1, {Φ¯ rf }3r=1}, {{φr}3r=1, g¯1, {Φ¯ rg}3r=1} be two sets of ...
We consider a class of multi-player games with perfect information and deterministic transitions, where each player controls exactly one non-absorbing state, and where rewards are zero for the non-absorbing states. With respect to the average reward, we provide a combinatorial proof that a subgame...
Every solution Π for the set of constraints Cons(P ), yields a predicated subtype hierarchy for P with which we can prove the safety of P . Theorem 1 [Soundness of Constraint Generation]. For every program P ≡ (T, Γ0, s), if Π is a solution for the constraints Cons(P ) then...